Uniform Asymptotic Expansions of the Incomplete Gamma Functions and the Incomplete Beta Function

Transcription

1 MATHEMATICS OF COMPUTATION, VOLUME 29, NUMBER 132 OCTOBER 1975, PAGES Uniform Asymptotic Expansions of the Incomplete Gamma Functions and the Incomplete Beta Function By N. M. Temme Abstract. New asymptotic expansions are derived for the incomplete gamma functions and the incomplete beta function. In each case the expansion contains the complementary error function and an asymptotic series. The expansions are uniformly valid with respect to certain domains of the parameters. 1. Introduction. The incomplete gamma functions are defined by (1 J) 'Y(a, X) = f: e-tta-ldt, f(a, X) = s: e-tta-l dt. The parameters may be complex; but here we suppose a and x to be real, where a > 0 and x ~ 0. Auxiliary functions are (1.2) P(a, x) = 'Y(a, x)/f(a), Q(a, x) = r(a, x)lr(a), and from the definitions it follows that (1.3) 'Y(a, x) + r(a, x) = r(a), P(a, x) + Q(a, x) = 1. For large values of x we have the well-known asymptotic expansion,. r(a, x) ~ xa-ie-x{l +(a - l)lx +(a - l)(a - 2)/x }. See for instance Dingle [1] or Olver [3]. If both x and a are large, this expansion is not useful, unless a = o(x). For large values of a, we can better use the function 'Y(a, x). From (1.1) we obtain the elementary result 'Y(a, x) = e-xxar(a) L xnlr(a + n + 1). n=o This series converges for every finite x. It is useful for a -+ oo and x = o(a), since under this condition the series has an asymptotic character. Expansions with a more uniform character are given by Tricomi [ 4 ), who found among others (1-4) 'Y(a + 1, a + y(2a)y 2 )/r(a + 1) = ~ erfc(-y) - ~(2/mr)Y2 (l + y 2 ) exp(- y 2 ) + O(a-1), y, a real, a This expansion is uniformly valid in y on compact intervals of R. The function erfc is the complementary error function defined by Received May 9, 1974; revised December 1, AMS (MOS) subject classifications (1970). Primary 33Al5, 41A60. Key words and phrases. Incomplete gamma function, incomplete beta function, asymptotic expansion, error function Copyright 1975, American Mathen,atical Society

2 1110 N. M. TEMME (1.5) It is a special case of r(a, x), namely erfc(x) = 7T-Y 2 f(\6, x 2 ). Some results of Tricomi are corrected and used by Kolbig [2] for the construction of approximations of the zeros of the incomplete gamma function '}'(a, x). An important book with many results on asymptotic expansions of the incomplete gamma functions is the recent treatise of Dingle [1 ]. Apart from elementary expansions, Dingle gives also uniform expansions and, in particular, he generalizes the results of Tricomi (p. 249 of [1 ]). Dingle does not specify the term "uniform", but it can be verified that the same restrictions on y must hold as for (1.4). In Section 2 we give new asymptotic expansions for '}'(a, x) and f(a, x), holding uniformly in 0 ~ xla for a and/or x In Section 3 an analogous result for the incomplete beta function is given. A recent result of Wong [5] may be connected with our results. Wong considers integrals of which the endpoint is near by a saddle point of the integrand, and he applies his methods to the function Sn(x) defined by n enx = L (nxylr! + (nxrsn(x)ln!. r=o The function Sn is a special case of the incomplete gamma function and the asymptotic expansion of Sn(x) for n , x ~ 1 is expressed by Wong in terms of the error function. (Wong interpreted his results only for 0 ~ x ~ 1, but not across the transition point at x = 1.) 2. Uniform Asymptotic Expansions. The integrals (1.1) are not attractive for deriving uniform expansions. Therefore, we write Pas (2.1) c > 0, in which (s + 1)-a will have its principal value which is real for s > - 1. Formula (2.1) can be found in Dingle's book. Here we derive it by observing that the Laplace transform of dp(a, x)ldx is (s + 1ra, from which it follows that (s + 1)-as- 1 = L(P(_a, )),which can be inve.rted to obtain (2.1 ). Taking into account the residue at s = 0, the contour in (2.1) can be shifted to the left of the origin; and so a similar integral for Q can be given. With (1.3) and some further modifications we arrive at (2.2) where Q(a, x) e-arp(a.) fc+i"" alj)(t)_!}!_.. e.., t, 1Tl c-1~ A - (2.3) If>( t) = t ln t, A.= xla. 0 < c <A., The contour in (2.2) will be deformed into a path in the s-plane which crosses the saddle point of the integrand. The saddle point t 0 follows from rp'(t 0 ) = 0. Hence t 0 = 1, rp(t 0 ) = rp'(t 0 ) = 0 and rp"(t 0 ) = 1. The steepest descent path follows from Im <P(t) = Im l/>(t 0 ) = 0, and, by writing t=a+it (a,rer)weobtain (2.4) a= r ctg r,

3 UNIFORM ASYMPTOTIC EXPANSIONS 1111 Let temporarily X > 1, that is x >a. Then the contour in (2.2) may be shifted into the contour L in the t-plane defined by (2.4). According to Cauchy's theorem, the integral in (2.2) remains unaltered; and on L, the values of </J(t) are real and negative. Next we define the mapping of the t-plane into the u-plane by the equation, (2.5) - ~u 2 = <P(t), with the condition t EL corresponds with u ER, and u < 0 if T < 0, u > O if r > 0. The result is Q( ) = e-a<p("a) f co -V.au2 dt ~ (2.6) a, x 2. e d ", 1fl -co U /\ - f The presence of the pole at t = :X in the integrand of (2.6) is somewhat disturbing, but we will get rid of it by writing (2.7) dt _1_ - dt _1_ + _1_ - _1_ du A. - t - du A. - t u - u 1 u - u 1 ' where u 1 is the point in the u-plane corresponding to the point t = :X in the t-plane. That is, - ~ui = <P(:X), hence u 1 = ± i<p(:x)'/ 2 There still is an ambiguity in the sign. However, the correct sign follows from the conditions imposed on the mapping defined in (2.5). In fact, we have (2.8) u 1 = i(l - X){2(A ln A.)/(1 - A.) 2 }v., where the square root is positive for positive values of the argument. The first two terms at the right-hand side of (2.7) constitute a regular function at t =_A., and with this partition we obtain e-a<p("a)f oo 2 du (2.9) Q(a, x) = -. e-v.au -- + R(a, x), 211"l -cc U - U1 (2.10) e-a<p("a)sco 2{dt 1 1 } R(a x) =. e-v.au du, ' 2m -cc du :X - t u - u 1 and the integral in (2.9) can be expressed in terms of the complementary error function defined in (1.5), so that (2.11) Q(a, x) = * erfc(av.n + R(a, x), s = iu 1 rv.. From erfc(x) + erfc(- x) = 2 it follows that (2.12) P(a, x) = * erfc(-av.n - R(a, x). So far, the results in (2.11) and (2.12) are exact, since no approximations were used. In order to obtain asymptotic expansions for P(a, x) and Q(a, x), the function R(a, x) will be expanded in an asymptotic series. The integrand of R(a, x) is a holomorphic function in the finite u-plane for every A.~ 0. If we put the expansion, dt = L: c c:x)uk, (2.13) du A. - t u - u 1 k=o k in (2.10), and, if we reverse the order of summation and integration, by Watson's lemma [3], we obtain the expansion

4 1112 N. M. TEMME (2.14) R(a, x) ~ ~ 2 -a<fi(a.) oo. L c2k0..)r(k + ~)(~a)-k-yz. 1Tl k=o (The conditions for Watson's lemma are certainly satisfied since the function is bounded for large real values of u.) Each coefficient ck("/i.) is an analytic function of A near A = 1, and the (2.14) is not only valid near )I,= I but for all A~ 0. That is to say, we car let a tend to infinity, or conversely. Also, x and a may grow dependently o dently of each other. The first few coefficients are Co(l) = -3, ("') - -i("/ "/I. + 1) - J_ ~A - ' 12("/I. - 1) 3 uf Our expansion is more powerful than those of Tricomi and Dingle. Tr mula (1.4) follows from our expansion by expanding (2.12) for small values Moreover, for the complete expansion, Tricomi and Dingle obtained an infini of which each term contains functions related to the error function. In our, the information about the nonuniform behavior of the incomplete gamma fu contained in just one error function. Besides, we obtain expansions for both Of course, the coefficients c2k("/i.) in (2.14) are more complicated than the C( in the other expansions. As remarked before, the expansion (2.15) is also valid for fixed a and.x spite of the nature of the series containing terms with negative powers of a. ficients, however, depend on x and a; and, in fact, we can say that the seq ue1 dk = c2k("/l.)a-k, is an asymptotic sequence. That is, dk+ 1 = o(dk) if one (c of the parameters a and x is (are) large uniformly in xla ~ The Incomplete Beta Function. The incomplete beta function is defil (3.1) I (p q) = - 1-fx tp- 1(1 - t)q-i dt x ' B(p, q) o with Re p > 0, Re q > 0; 0 ~ x ~ 1, and (3.2) B(p, q) = r(p)r(q)/r(p + q). The function in (3.2) is called the beta function. Again, we consider real vari p and q, and we will derive an asymptotic expansion of Jx(P, q) for large pa formly valid for 0 < o ~ x ~ I. We first give an integral representation of Ix which resembles those for plete gamma function. Formula (3.1) is equivalent to (3.3) I (p, q) = _l_f oo and also, we have (3.4) x B(p, q) -In x i e-pt(l - e-t)q-1 dt;

5 UNIFORM ASYMPTOTIC EXPANSIONS 1113 from which follows, by using the same technique as in the foregoing section, I (p q) =. 1 sc+ioo e<p-s)ln x ~ ds X ' 21TiB(p, q} c-ioo p - S ' This expression can be written as 0 <c <p. (3.5) with i/j(t) = t ln(t/x) - (1 + t) ln(l + t) - ln(l - x), (3.6) F(t)= r(qt)eqt(qttqt t 1 =plq,ando<c<t 1 q r(q + qt)eq(l+t)(q + qttq(l+t)' Fq(t) is a slowly varying function as q-+ co, on compact subsets of larg tl < 1T, t =I=- 0. Of course, its construction is based on the Stirling approximation of the gamma function. For larg tl < 1T, t =I=- 0, we have (3.7) Fq(t) = {(1 + t)/t}v.(1 + O(q-1)), q-+ co, t fixed. With (3.8) t 0 = x/(1 + x), x 0 = pl(p + q), t 0 is a saddle point of 1/1, and if x = x 0, this saddle point coincides with the pole at t 1 The calculation of the saddle point t 0 is based on the assumption that the gamma functions in Fq(t) in (3.6) have large arguments. Hence, for small values of x, which correspond to small values of t 0, the calculation is based on false assumptions. Therefore we only consider positive values of x. It is not necessary to have a: uniform bound from zero of x. We are even allowing those values of x with qx From now on, details will be omitted, since the method is exactly the same as the one used in the foregoing section. We put - ~u 2 = t/j(t) and the results are (3.9) (3.10) Ix(P, q) = ~ erfc(-(q/2)%11) + Sx(P, q ), 1/ = (x - x 0 )[2q- 1 {p ln(x 0 /x) + q ln((l - x 0 )1(I - x))}l(x - x 0 ) 2 ]v.. The square root is positive for positive values of its argument. The function S x is defined by (3.11) S ( ) = xp(i - x)qr(q)e-qqq J.. -Y.qu2G( )d X p, q 2 "B( ) e u u, 1Tl p, q -oo (3.12) Fq(t) dt Fq(t 1 ) G(u)= t 1 - t du u - u 1 For u 1 we have - ~ui = l/l(t 1 ), u 1 = iri. The role of the parameter "A. of the foregoing section is now played by (x - x 0 ). We have " ~ -7(P; q )"(x - x 0 + : P';. q' (x - x 0 ) + O(x - x 0)'1

6 1114 N.M.TEMME for x--? x 0. An asymptotic expansion of Sx is obtained by expanding G(u) = L dkuk, giving (p q) - - eqq-q L., dzkf'(k + ~)(~q)-k-yz' (3.13) S - xp(l -x)q f'(p + q) ~ x ' 21Ti f'(p) k=o (3.14). Fq(t 0 ) _x'lz _ Fq{t1 ) d =1-~o t 1 -t 0 1-x u 1 The expansion holds for p --? 00 and/or q --? 00, unifonnly in o < x < l, where o may depend on q, such that qo --? 00 A more transparent first approximation for Sx(p, q) is obtained by replacing the functions Fq in (3.14) by the approximation (3.7). The result is Sip, q) = {p/[21tq(p + q)]}y 2 (x/x0)p {(I - x)/(1 - x 0)}q q--? oo. Department of Applied Mathematics Mathematical Centre Amsterdam, The Netherlands 1. R. B. DINGLE, Asymptotic Expansions: Their Derivation and Interpretation, Academic Press, London and New York, K. S. KOLBIG, "On the zeros of the incomplete gamma function," Math. Comp., v. 26, 1972, pp MR 48 #5336. York, F. W. J. OLVER, Asymptotics and Special Functions, Academic Press, London and New 4. F. G. TRICOMI, "Asymptotische Eigenschaften der unvollstandigen Gammafunktion," Math. Z., v. 53, 1950, pp MR 13, 553. S. R. WONG, "On uniform asymptotic expansion of definite integrals," J. Approximation Theory, v. 7, 1973, pp